A290316 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 - 21\,x - 57\,x^{2} - 47\,x^{3} - 12\,x^{4} + 15\,n x + 36\,n x^{2} + 27\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(3\,x + 3\,x^{2} - 9\,x^{3} - 15\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	7														8
3	1	50	139													190
4	1	237	2685	4609												7532
5	1	1000	30426	172444	214297											418168
6	1	4067	277006	3598426	13152725	12820999										29853224
7	1	16350	2260365	57134324	438656631	1169007846	937923283									2604998800
8	1	65497	17359491	777526955	10784605475	57018357003	118943065321	81109986433								268650966176
9	1	262100	128655652	9617733488	220805932270	2009414906468	7984821768940	13649288125736	8094667157329							31968744541984
10	1	1048527	933090012	111755676972	4010001274566	57831852719178	382065425508756	1206799379897076	1745009081488521	915646445959303						4311474876662912
11	1	4194250	6677375391	1244507340648	67032711387138	1450633013571420	14724188418238950	75313663586081688	196525129663916829	246030764661270874	115769395774250971					649882059017628160
12	1	16777157	47387542297	13453583580021	1055932848467466	32982140676584946	488162645483618658	3747081779436153786	15509167011342765093	34382349600365993113	37931152230933995669	16178897851297498561				108270862693372976768
13	1	67108800	334548135678	142403174450932	15922620080904519	697309278591994704	14497635530048224596	158800203616429714680	968183403342936908751	3347992469195444731792	6440471417416192638798	6348894338941134253428	2476453714715956273993			19756006557394605340672
14	1	268435387	2354184969018	1484589386247838	232361898992165323	13951096743050556417	396156556752622773324	5972991873840467457252	51087333762982491441111	256321689990864060056989	758569810691403614699482	1287280284503858814080094	1146644172650419368831229	412037876955756202893511		3918324501931463524606976
15	1	1073741750	16532713425217	15307333443753772	3307385814651937121	267455384060495184322	10148377211795863670577	204968280310639301059176	2374933448623297052269995	16485580437013333986574522	69904579892724197308890347	180393257671833358848401836	273630459061662921290882035	222271811152246169110685582	74042038979690777284629667	839318048079410232425105920

Technical details

Node id: n-e52b4819ab506a80 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-f90db304c0a6f319

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,       7;
   3: 1,      50,       139;
   4: 1,     237,      2685,         4609;
   5: 1,    1000,     30426,       172444,        214297;
   6: 1,    4067,    277006,      3598426,      13152725,       12820999;
   7: 1,   16350,   2260365,     57134324,     438656631,     1169007846,       937923283;
   8: 1,   65497,  17359491,    777526955,   10784605475,    57018357003,    118943065321,      81109986433;
   9: 1,  262100, 128655652,   9617733488,  220805932270,  2009414906468,   7984821768940,   13649288125736,    8094667157329;
  10: 1, 1048527, 933090012, 111755676972, 4010001274566, 57831852719178, 382065425508756, 1206799379897076, 1745009081488521, 915646445959303;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(15*n-21)*x+(36*n-57)*x^2+(27*n-47)*x^3+(6*n-12)*x^4)*P(n-1,x) + (3*x+3*x^2-9*x^3-15*x^4-6*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (15*n - 21)*x + (36*n - 57)*x^2 + (27*n - 47)*x^3 + (6*n - 12)*x^4)*P[n - 1, x] + (3*x + 3*x^2 - 9*x^3 - 15*x^4 - 6*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + (15*n - 21)*x + (36*n - 57)*x^2 + (27*n - 47)*x^3 + (6*n - 12)*x^4)*v[i - 1] + (3*x + 3*x^2 - 9*x^3 - 15*x^4 - 6*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003